Comment by meroes
10 months ago
It’s a simplification that all kinds of infinites are implicated by infinite models? To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC?
Why not pick 10^180 or something as the highest number/states then?
Yes even natural numbers are infinite, so I don’t quite buy at face value it’s a simplification when we could cutoff these infinities because no computation will run 10^180 for attoseconds or have that many states. And I would think all the mysteries of ZFC for example being ousted would be simpler.
I think for your simplification it may have been clearer to say abstraction.
> Why not pick 10^180 or something as the highest number/states then?
Because then we have to find all sorts of limits to all of our proofs, and qualify everything, and waste time proving that our results are under this limit? If we prove something about an algorithm parameterized by some N, and we rely on a Turing Machine in our proof, then we'd also have to figure out which N hits our limit of 10^180 states on the tape, so we can qualify the limits of our proof. This could be very difficult, since the translation to a Turing machine model can be arbitrarily complex. Why bother? It doesn't help anyone.
> To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC?
Turing machines are for proving things about the nature of computation itself. You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation. That's simply not true. If Turing machines help you prove something, use them; if not, ignore them. Putting an arbitrary limit on them only makes them worse at proving things.
I'm not really sure why you're so focused on ZFC. On one hand, the vast majority of math proofs (and therefore CS proofs that are based on them) assume ZFC, usually without even bothering to mention it. On the other hand, the axiom of choice seems completely irrelevant for Turing machines. Although the tape is infinite, it's almost always implicitly assumed that the state is finite at every step (the rest of the tape is 0). You'd have to initialize the tape with some infinite pattern to get around this, and that's definitely out of the ordinary. It's probably better to think of the state as "arbitrarily large" rather than infinite. I suspect you could get away with assuming Peano arithmetic for most CS proofs anyone cares about.
But also, yes, some number of scientists and mathematicians are expected to understand the basic axioms of math if they're going to prove mathematical theorems.
> You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation.
All computations are finite, like you said. In general it’s a mystery to me how infinites of math are indispensable to the best sciences. In a sense it is simple, sure-we just use it when it helps us like you said. But since the indispensability of modern math is still hotly talked about in philosophy, I don’t know that I would call the finitely bound physical using infinity, simple. Do I think the difficulties of a mathematical and computational system with a biggest number (one we can pluck from cosmology as the largest number of useable states before total heat death) are simpler than one that uses infinite infinities at the lowest level, spawning philosophical challenges? I don’t know. I just can’t confidently call the status quo simple.
Computations are finite, but that limit keeps going up. If we picked a number that seemed like "way more than enough" in 1960, it might be less than our phones can do now. If we picked a number from cosmology, what happens if that science changes? "Heat death" is the current consensus of our distant future, but even that is challenged. Until we know what dark energy is, we can't be sure about "heat death". So if cosmology has some breakthrough that changes our understanding, why make it so we have to go revisit all our CS proofs? Using infinity decouples the proofs from irrelevant externalities.
> the indispensability of modern math is still hotly talked about in philosophy
Meh, philosophers have to talk about something. I'll worry about what philosophers say when I see a single useful insight from the entire field. We've been waiting thousands of years, but I'm sure a useful thought is right around the corner.
And the concept of infinity is hardly modern.